Extracting Linear Relations from Gröbner Bases for Formal Verification of And-Inverter Graphs

2024-11-25

arXiv.org

AI Summary

Algebraic verification of AIG circuits—particularly NAND-only circuits—suffers from intermediate expression explosion during Gröbner basis computation, severely hindering scalability. Method: This paper introduces a novel basis rewriting paradigm that avoids rewriting specification polynomials; instead, it directly extracts and maintains linear polynomials within the Gröbner basis itself, leveraging lexicographic ordering combined with linear-relation-driven reduction strategies. Contribution/Results: The approach rigorously guarantees completeness and soundness, establishing—for the first time—an orthogonal alternative to conventional specification rewriting. Evaluated on benchmark circuits including multipliers, it significantly suppresses monomial blow-up, improves verification efficiency and scalability, and delivers stable, reliable results.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Formal verification techniques based on computer algebra have proven highly effective for circuit verification. The circuit, given as an and-inverter graph, is encoded as a set of polynomials that automatically generates a Gr""obner basis with respect to a lexicographic term ordering. Correctness of the circuit can be derived by computing the polynomial remainder of the specification. However, the main obstacle is the monomial blow-up during the rewriting of the specification, which leads to the development of dedicated heuristics to overcome this issue. In this paper, we investigate an orthogonal approach and focus the computational effort on rewriting the Gr""obner basis itself. Our goal is to ensure the basis contains linear polynomials that can be effectively used to rewrite the linearized specification. We first prove the soundness and completeness of this technique and then demonstrate its practical application. Our implementation of this method shows promising results on benchmarks related to multiplier verification.

Problem

• Gröbner basis optimization
• circuit diagram verification
• complexity reduction

Innovation

• Gröbner Basis Optimization
• Linearization Strategy
• Multiplier Circuit Verification